Irreducible projective representations of the symmetric group which remain irreducible in characteristic 2

For any finite group G and any prime p one can ask which ordinary irreducible representations remain irreducible in characteristic p. We answer this question for p=2 when G is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan-Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.